The blades of a windmill turn on an axis that is 30 feet from the ground. The blades are 10 feet long and complete 2 rotations every minute.

Write a sine model, [tex]\(y = a \sin(b t) + k\)[/tex], for the height (in feet) of the end of one blade as a function of time [tex]\(t\)[/tex] (in seconds). Assume the blade is pointing to the right when [tex]\(t = 0\)[/tex] and that the windmill turns counterclockwise at a constant rate.

A. [tex]\(y = 30 \sin \left(\frac{\pi}{15} t\right) + 10\)[/tex]

B. [tex]\(y = 30 \sin \left(\frac{\pi}{15} t\right) + 30\)[/tex]

C. [tex]\(y = 10 \sin \left(\frac{\pi}{15} t\right) + 10\)[/tex]

D. [tex]\(y = 10 \sin \left(\frac{\pi}{15} t\right) + 30\)[/tex]



Answer :

To write a sine model for the height of the end of one blade as a function of time, we need to analyze the given parameters systematically.

### 1. Understanding the Axis and Blade Height
- The axis of the windmill is located 30 feet above the ground.
- The blades extend 10 feet from the axis.

### 2. Defining Amplitude ([tex]\(a\)[/tex])
- The amplitude of the sine function is the maximum deviation from the center value, which corresponds to the length of the blade.
- Therefore, [tex]\(a = 10\)[/tex] feet.

### 3. Defining Vertical Shift ([tex]\(k\)[/tex])
- The vertical shift represents the height of the axis above the ground because that is the center line about which the blade oscillates.
- Therefore, [tex]\(k = 30\)[/tex] feet.

### 4. Defining Period and Frequency ([tex]\(b\)[/tex])
- The windmill completes 2 rotations every minute.
- Thus, 1 rotation takes [tex]\(30\)[/tex] seconds.
- The period of the sine function, which corresponds to one full rotation, is [tex]\(30\)[/tex] seconds.

Frequency ([tex]\(b\)[/tex]) can be calculated from the period using the formula:
[tex]\[ b = \frac{2\pi}{\text{period}} \][/tex]
[tex]\[ b = \frac{2\pi}{30} \][/tex]
[tex]\[ b = \frac{\pi}{15} \][/tex]

### 5. Writing the Sine Function
Putting it all together, the height [tex]\(y\)[/tex] of the end of one blade as a function of time [tex]\(t\)[/tex] can be modeled by:
[tex]\[ y = a \sin(bt) + k \][/tex]
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]

Thus, the correct sine model is:
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]

Out of the provided options, the correct one is:
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]